MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
String Theory (8.821) – Prof. J. McGreevy – Fall 2007
Solution Set 8
Worldsheet perspective on CY compactification
Due: Monday, December 18, 2007 at 11:00 AM in the box or in my office.
1. Orbifolds.
Suppose you have in your possession a 2d CFT with the correct central charge
to provide (part of) a critical string background. Suppose further that this
CFT has a symmetry group Γ (for simplicity let’s assume it’s discrete, finite
and abelian). For example, say the CFT is a sigma model, and the target space
has a discrete isometry, which could be a subgroup of a continuous isometry.
Let’s try to make a new CFT by gauging this symmetry; in particular, this will
mean projecting onto states invariant under Γ.
(a) By considering the torus amplitude of this theory, show that a modular
transformation (τ → −1/τ ) relates the projection onto Γ-invariant states to
a sum over new sectors of states called twisted sectors. These are sectors of
strings which are only closed modulo the action of Γ:
x(σ + 2π) = γx(σ)
for some γ ∈ Γ.
The resulting procedure of projecting onto invariant states and summing over
twisted sectors in a modular-invariant way is called the orbifold construction.
Projecting onto invariant states in the torus vacuum amplitude is
accomplished by inserting into the trace the projection operator
X
P =
Uγ
γ∈Γ
where Uγ is the unitary operator that implements the orbifold action
on the string hilbert space. Consider a particular term in this sum:
it computes
X
hn|Uγ e−τ2 H+iτ1 P |ni,
tr Uγ e−τ2 H+iτ1 P =
n∈Huntw
1
where the sum over states in the second line is over the untwisted
sector of closed string states quantized on a circle of fixed (euclidean)
worldsheet time σ2 ; this sector to be distinguished from the twisted
sectors we are about to discover. A modular transformation τ → − τ1
interchanges what we consider the space and time dimensions on the
worldsheet. Hence it exchanges the circle of constant time on which
we construct the hilbert space, with the euclidean time circle that
implements the trace. In the dual channel, then we are inserting the
operator Uγ at a point in the spatial circle. This is an instruction to
act with the orbifold generator γ when going around the loop of the
string – the string only needs to close up to the action of gamma. We
see that modular invariance of the torus vacuum amplitude demands
that we include such sectors.
If our worldsheet theory can be described in terms of some fields X,
we can restate this more simply. The τ → − τ1 modular transformation
relates the sector where
X(σ1 + 2π, σ2 ) = X(σ1 , σ2 ), X(σ1 , σ2 + 2π) = γX(σ1 , σ2 )
to the one where
X(σ1 + 2π, σ2 ) = γX(σ1 , σ2 ), X(σ1 , σ2 + 2π) = X(σ1 , σ2 ).
The inclusion of such sectors is very natural if we think of the orbifold
group as a (discrete) gauge symmetry – configurations related by
the action of the gauge group are simply the same, and so on the
configuration space of the orbifolded theory, strings that only close
up to the action of Γ must be included.
(b) Consider the simple example where our initial CFT is four free bosons of
infinite radius; group them into complex pairs z i=1,2 , and consider the group
Γ = ZZN generated by
1 1
z
ω
0
z
, ω N = 1.
→
γ:
−1
2
z2
0 ω
z
Show that putting a type II string on this CFT preserves half the supersymmetry. (This is a model of part of a K3 surface.)
For the case of N = 2, compute the massless spectrum. Note that γ does not
act freely; the geometric space you get by identifying points under gamma is
2
therefore scary to mathematicians; it is called an AN −1 singularity. Strings
propagation on the other hand is completely fine.1
γ is a rotation matrix that lies in SU(2) ⊂ SO(4). The action of γ
defines an action on spinors; this SU(2) subgroup preserves a spinor
of SO(4) ∼ SU(2) × SU(2), since the (Dirac) spinor rep is (2, 1) ⊕ (1, 2).
Once we know that there are 16 supercharges preserved, it is easier
to compute the spectrum since it must lie in multiplets of the supersymmetry. Depending on whether we start with type IIA or IIB, we
get 6d N = (1, 1) or (2, 0) supersymmetry (the two numbers are the
numbers of left-handed and right-handed supercharges respectively).
Let’s focus on the IIA case. In this case, the twisted sector contains
a massless (1, 1) vectormultiplet.
If you want more details about this, ask me.
(c) [BONUS] Note that the orbifolded theory has a new global discrete symmetry, called the quantum symmetry (not my fault), which acts with a phase
αk on the sector twisted by ω k . This symmetry constrains the interactions of
the orbifold theory. What can you say about the theory you get by orbifolding
by the quantum symmetry?
You get back the original CFT before orbifolding. The states invariant under the quantum symmetry are exactly the untwisted states.
The twisted sectors of the orbifold by the quantum symmetry are
the states that weren’t invariant under the original projection.
2. GLSM (Gauged Linear Sigma Model).
A powerful technique for studying nonlinear sigma models on CY manifolds can
be found by studying massive (meaning not scale-invariant) 2d QFTs which
RG flow to them in the IR. In particular, the parameter space of the massive
theories will give some coordinates on the moduli space of the CFT in the IR.2
Sigma models with Kähler target have 2d (2, 2) supersymmetry. So to find good
candidate models, let’s look at the multiplets of (2, 2). The most accessible are
1
To be precise, it can be shown that the orbifold CFT has a certain nonzero NSNS Bmn in the
background which resolves the singularity. By deforming this to zero it is actually possible to make
the perturbative string theory singular. In that limit some wrapped D2-branes become light and
can enhance the gauge symmetry (the abelian gauge symmetry coming from the RR vectors) to
SU (N ).
2
The reference for this discussion is E. Witten, “Phases of N = 2 theories in two dimensions.”
hep-th/9301042.
3
are
(i) the chiral multiplet, D̄α Φ = 0, which can be parametrized as
Φ(y, θ) = φ(y) + θα ψα + θα θα F
where φ is complex, α = ±, and F is an auxiliary field.
(ii) the vector multiplet, V = V̄ , which contains a vector, gaugino and auxiliary
field D. It can be described in terms of a twisted chiral field Σ ≡ D̄+ D− V ,
satisfying D̄+ Σ = 0 = D− Σ. Its expansion is
Σ = σ + θ+ λ̄+ + θ̄− λ− + θ+ θ̄− (D + iv01 ) + ...
where v01 is the field strength of the U(1) gauge field.
In two dimensions, even abelian gauge theories confine – the maxwell term is
by dimensional analysis a relevant operator. Consider a U(1) gauge symmetry,
with N + 1 chiral multiplets of charge 1. An important term in the lagrangian
will be the Fayet-Ilioupoulos (FI) term; this is a term
Z
= dθ̄+ dθ− Σt = −rD + θv01
with t = r + iθ
The whole bosonic potential so far is
LD =
X
D2
+
D
|φi|2 − Dr.
2e2
(a) Integrate out the auxiliary field D to find the bosonic potential. Show that
when r is large and positive (we will denote this regime by the awkward symbol
‘r ≫ 0’), the φs can’t simultaneously vanish and keep zero energy. Convince
yourself that the vacuum manifold is therefore CIPN . It is then reasonable to
conjecture that this gauge theory flows to a nonlinear sigma model on CIPN
with a size related to the FI parameter.
The terms in the lagrangian involving the D-term are
N +1
X
D2
LD = 2 + D
|φi |2 − Dr.
2e
i=1
4
The equation of motion for D is algebraic:
N
+1
X
δL
=⇒ D = −e2
0=
δD
i=1
!
|φi|2 − r .
Plugging this back into LD gives
N
+1
X
e2
LD | = −
2
i=1
!
|φi |2 − r .
When r ≫ 0, zero D-term energy then requires that not all of the φs
vanish. The vacuum manifold is
N
CIP = {
N
+1
X
i=1
|φi |2 = r > 0}/U(1) = {φi} − {0} /C⋆ .
(b) Show that the beta function for the FI parameter is proportional to the
sum of the charges of the chiral fields. Relate this to your expectations about
the RG flow of the CIPN model.
The FI coupling appears in the lagrangian as the coupling constant
for a single D field:
rD.
It appears in other terms related to this one by supersymetry. A
term which can generate corrections to this one is
qi |φi |2 D
(where qi is the U(1) charge of φi since the contraction of a pair of φs
contains a c-number piece which will renormalize r. The propagator
for φ in momentum space is
p2
1
;
+ m2φ
the mass of the φ field is qi2 σ 2 . So the renormalization of r at one
loop is
X Z Λ d2 p
1
δr =
qi
2
2
p p + m2φi
i
5
Introducing a UV cutoff Λ, this is
δr ∝
X
qi ln
i
Λ
.
m2φ
The beta function is then
βr ∼ Λ
X
d
δr ∝
qi .
dΛ
i
When the sum of the charges is zero, the FI parameter does not run.
We will see below that this is the same as the CY condition on the
geometry.
Actually, the renormalization of r can be argued to be one-loop exact,
since it’s related by supersymmetry to the anomaly in the axial Rsymmetry.
(c) To get a CY threefold, take N = 5 above, and add one more field P of
charge −5. Note that the FI parameter no longer runs. Now we can add a
gauge-invariant superpotential
W = P G5 (Φ).
Find the bosonic potential. In the case where G is transverse, show that the
vacuum manifold is {P = 0} and
{G5 (φ)} ∈ CIP4 .
The fluctuations perpendicular to this locus are all massive, and the IR limit
seems to give a sigma model on the quintic.
The bosonic potential is UD + UF with
e2
UD = −
2
and
UF =
X
chirals,f
|
−5|p|2 +
N
+1
X
i=1
!
|φi|2 − r .
X ∂W 2
∂W 2
| = |G5 (φ)|2 + |p|2
| i| .
∂f
∂φ
i
For r ≫ 0, the D-term potential can only vanish if at least one of
the φs is nonzero. If G is transverse (meaning G = dG = 0 has no
6
solutions other than φi = 0, ∀i) then the second term in the F-term
potential forces P to vanish. Then the first term (which much vanish
independently of the second since they are both positive) forces the
massless modes onto the locus G5 = 0 which is the quintic.
Relate the moduli of the quintic to the marginal parameters of the gauge theory.
Note that the word ‘moduli’ is a bit too useful in string theory. The
word ‘moduli’ here doesn’t refer to the vacuum manifold of the 2d
gauge theory (which could be called a ’moduli space’ if not for the
Coleman-Mermin-Wagner-Hohenberg theorem) . Rather, it refers to
deformation parameters determining the geometry of the CY. (These
are moduli of the 4d effective theory of strings compactified on the
CY.) They correspond (at least some of them) to parameters of the
2d gauge theory that don’t run under the RG flow.
In this example, these parameters are the (complexified) FI parameter and the parameters in the superpotential. From part (a) we know
that the FI parameter controls the size of the projective space into
which we’re embedding the quintic, so it controls the kähler class
of the CY. The parameters in the superpotential of the 2d gauge
theory determine the complex structure moduli of the quintic CY
target space of the NLSM to which the gauge theory flows. There
are 126 degree-five monomials in 5 variables that could appear in
the superpotential; you can get rid of 25 of the coefficients by field
redefinitions. This counting is just the same as when we counted
complex structure moduli of the quintic hypersurface.
(d) Consider varying the complex structure moduli (via the coefficients of G5 )
so that G5 is not transverse, i.e. where G = dG = 0 does have a solution for φ
not all zero. What happens to the vacuum manifold in this case?
If G is not transverse, i.e. there are values of φi = φi⋆ 6= (0, .., 0) which
solves G = dG = 0, then p is not forced to vanish when φi = φi⋆ . That
is, the vacuum manifold has a branch structure and is not a manifold.
Even worse, the extra branch whose coordinate is p is not compact,
unlike the quintic. The path integral then contains an extra integral
over the zero mode of p which will make some correlators diverge.
This GLSM can be expected to flow to a singular CFT.
(e) There’s another semiclassical limit where |r| is large, namely r ≪ 0. Convince yourself that the D term condition implies that P must have an ex7
pectation value in this limit. Show that the resulting effective theory is a
Landau-Ginzburg orbifold, a model of massless fields with a superpotential
W ∝ G5 (φ)
orbifolded by a ZZ5 symmetry. This is a completely non-geometric CFT, connected by variation of Kähler moduli to the quintic NLSM.
Now the D-term condition forces p not to vanish. We can solve the
D-term for p and use the U(1) to pick the phase of p so that it’s
positive:
√
hpi = + r.
The second term in the F-term potential then forces (for transverse
G) all the φs to have vanishing vevs. Although the φs have vanishing vevs, they do not have mass terms. Such a model where massless fluctuations are governed by a higher-order potential is called a
Landau-Ginzburg theory. The remaining superpotential is
√
W = P G5 (φ) = rG5 (φ),
where we have set P to its vev which we can do because its fluctuations are massive.
There is just one remaining subtlety. Since P has charge −5, its vev
does not completely break the gauge symmetry. Gauge transformations
φ → ωφ, P → ω −5 P
with ω 5 = 1 do not change the vev of P and hence still act on the
φ fluctuations. This discrete gauge symmetry implements a ZZ5 orbifold.
(f) Remember that the vectormultiplet contains a complex scalar σ. Under the
relation to 4d N = 1, these modes related to the components of the 4d gauge
field along the two dimensions along which we reduced. This tells us that there
must be potential terms of the form
!
X
|φi |2 + 25|p|2 .
V ⊃ |σ|2
i
When r ≫ 0,
|φi |2 ∼ r means that σ gets a large mass; similarly when
r ≪ 0, |p|2 ∼ r gives σ a large mass. What happens in between? Naively at
P
8
r = 0, we can set φi = p = 0 consistent with D = 0, dW = 0, so it would seem
that σ becomes a flat direction there! Such an extra branch of the target space
of the sigma model at a point in its parameter space would lead the correlators
of the CFT to diverge there. 3
The expectation that there should be a singularity where σ becomes a
flat direction is correct. However, since the parameters of the GLSM
must describe vevs of fields in a supersymmetric 4d effective theory,
we know they must come in complex
R pairs. The complex partner of
the FI parameter is the theta angle θv01 , as indicated before part (a)
of this problem. When θ 6= 0, it forces there to be a nonzero electric
field in the vacuum on the Coulomb branch (the ’Coulomb branch’
refers to possible vacau where the φs (which are charged under the
U(1) and hence Higgs fields) are zero, but σ which is neutral can
have a vev). This electric field costs energy and lifts the Coulomb
branch. Hence there is just one special value of r, θ where a new
noncompact σ branch sticks out of our vacuum manifold.
(g) We saw above in section (d) how to make a GLSM for an orbifold. Consider
the GLSM with one U(1) and three chiral fields (Z 1 , Z 2 , P ) of charges (1, N −
1, −N). Show that the FI parameter is marginal. Show that when r ≪ 0, this
flows to the orbifold of C2 studied in problem 1. Interpret the variation of the
FI parameter as condensing a mode of the twisted sector (this is called blowing
up the singularity). In the case N = 2, show that in the space at r ≫ 0, the
orbifold point has been replaced by a IP1 . It is this 2-sphere that shrinks in
making the A1 singularity, and this is why a D2-brane can become massless
there.
The sum of the charges is 1 + N − 1 − N = 0 so the FI parameter
does not run. When r ≪ 0, the vev of P is forced to be nonzero
by the D-term, and breaks the gauge symmetry down to ZZN . This
ZZN acts on Z 1,2 exactly as in problem 1c). The FI parameter can be
interpreted as the vev of a twisted sector field because it acts as a
chemical potential for vortices of the gauge field.
(i) [BONUS] Compute the Witten index of the gauge theory with N + 1 chiral
fields based on our answer for the vacuum manifold at large radius. This means
that there are at least N + 1 supersymmetric vacua. What happens to them
in the IR when the CIPN shrinks to nothing?
3
Actually the correlators only diverge if the theta angle is also zero, because otherwise there is
a constant electric field in the vacuum.
9
Recall that in the calculation of the beta function for the FI parameter,
the mass of the φ fields depended on σ. In the case where
P
q
=
6
0,
the running of the FI parameter can be thought of as an
i i
effective potential for σ. In a 2d (2, 2) theory, such a potential comes
from a twisted chiral superpotential:
Z
d2 θ̃W̃ (Σ)
R
where d2 θ̃ ≡ dθ+ dθ̄− is an integral over a different half of superspace
(the half that doesn’t kill a twisted chiral superfield D̄− Σ = 0 =
D+ Σ), and Σ ≡ D̄− D+ V is a twisted chiral superfield made from the
vectormultiplet whose lowest component is σ. The twisted chiral
superpotential which encodes the running of the FI parameter is
X
W̃ (Σ) = tΣ −
qi Σ ln Σ/Λ;
i
the first term is just the tree-level FI parameter.
P
For P N , i qi = N + 1. The condition on Σ for there to be a supersymmetric vacuum is
0=
∂ W̃
= t − (N + 1) ln Σ/Λ + N + 1
∂σ
which is solved by
hσi ∼ Λet/N +1 ωN +1
where ωN +1 is an N + 1th root of unity, of which there are N + 1. This
is where the vacua go – the squirt out onto the sigma branch.
10